First slide

Permutations

Question

The number of 7 digit numbers the sum of whose digits is even, is

Moderate

A

$35 \times 10^{5}$
B

$45 \times 10^{5}$
C

$50 \times 10^{5}$
D

None of these

Solution

A number of the seven digits will be of the form.
${ax}_{1} x_{2} x_{3} x_{4} x_{5} 0, {ax}_{1} x_{2} x_{3} x_{4} x_{5} 1, {ax}_{1} x_{2} x_{3} x_{4} x_{5} 2, {ax}_{1} x_{2} x_{3} x_{4} x_{5} 3, \dots {ax}_{1} x_{2} x_{3} x_{4} x_{5} 9$
where $a \in {1, 2, 3, \dots 9}$
and $x_{1}, x_{2}, x_{3}, x_{4}, x_{5} \in {0, 1, 2, 3, \dots 9}$ .
Since sum of the digits should be even, therefore,
if $a + x_{1} + x_{2} + x_{3} + x_{4} + x_{5}$ is an even number, then the digit at units place must be 0, 2, 4, 6, 8 and if $a + x_{1} + x_{2} + x_{3} + x_{4} + x_{5}$ is an odd number, then the digit at units place will be 1, 3, 5, 7, 9.
∴ Required number = 9 × 10 × 10 × 10 × 10 × 10 × 5
= 9 × 10⁵ × 5 = 45 × 10⁵.

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